About the project

I am so excited to get an opportunity to be into R. I am expecting to learn how to hand data and visulaze the results of analysis through R. One of my colleagues had let me know about this course.

The URL for my git repository as follow:

https://github.com/Donggyu-Kam/IODS-project


Regression and Model validation (Excercise 2)

The data is the student’s attitude toward statistics.The data consists of information about students’ ages, students’ gender, three approaches, students’ attitude toward statistics and exam points. The oberservations whare the exam points are zeor were removed from data. And thus, the data has 166 observations and 7 variables. You can figure out three variables to affect the exam points through linear regression models and finally you will figure out the most significant variable to exampoints

#Before handling data, you should install two packages:“ggplot2” and“GGally”.

1.To read and explore data

  • Read data and look into the structure and dimension of data
  • Initialize plot with data and aesthetic mapping
    • p1 <- ggplot(df, aes(x = attitude,col=gender, y = points))
  • Define the visualization type (points)
    • p2 <- p1 + geom_point()
  • Draw the plot
    • p2
  • Add a regression line
    • p3 <- p2 + geom_smooth(method = “lm”)
  • Add a main title and draw the plot
    • p4 <- p3+ggtitle(“<Student’s attitude versus exam points>”) &
    • p4

2. To choose variables and create a regression model

By drawing plots illustrating the relationships between the variables and the normality of the erros distributions, it will find out what variables are closely related to exam points.

  • Draw a scatter plot matrix of the variables in learning2014
    • p <- ggpairs(df, mapping = aes(col=gender, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20)))
  • Plotting
    • p

According to the plot “p”, three variables “attitude”, “stra” and “surf” were chosen due to the strongest correlation value as explanatory variables.

  • Fit a linear model to each variable
    • qplot(attitude, points, data = df) + geom_smooth(method = “lm”)
    • qplot(stra, points, data=df)+ geom_smooth(method =“lm”)
    • qplot(surf, points, data=df)+ geom_smooth(method=“lm”)

After fitting each variable to a linear regression modle, “attitude” shows postive relationship but the others show week or negative relationships. So, the linear regression model is re-fitted with three variables together.

  • Fit a linear modewl with three variables together
    • mymodel<-lm(points~attitude+stra+surf, data=df)
  • Take a look at the summary of the model
    • summary(mymodel)

In this model, “Stra” and “Surf” are not not significantly related to the exam points. So, the model is re-fitted again witout the two variables

  • Fit the model again with the “attitude”

    • mymodel2<- lm(points ~ attitude, data=df)
    • summary(mymodel2)
    • par(mfrow = c(2,2))
    • plot(mymodel2, which=c(1,2,5)) - only take looks at the relation between residuals and fitted values, the normal QQplot, and the leverage of observations.

3. Result of regression model analysis

By analyzing the residuals and errors of the model, we can test the model validation. The variance of errors in residulas over fitted values looks constant without any patterns but there are 4 outliers indicating that the model only with “attitude” cannot explain the 4 ouliters of exam points. In the normality of errors in QQ plot, the errors are normaly distributed. In leverage of observations, standardized residuals are scattered around 0 and there seems no impact of outliers


Losistic Regression (Excercise 3)

The data is about student acheivement in secondary education of two Portugues schools. Two datasets are available for the performance in two subjects: Math and Portuguese language.

P. Cortez and A. Silva. Using Data Mining to Predict Secondary School Student Performance. In A. Brito and J. Teixeira Eds., Proceedings of 5th FUture BUsiness TEChnology Conference (FUBUTEC 2008) pp. 5-12, Porto, Portugal, April, 2008, EUROSIS, ISBN 978-9077381-39-7.[link]http://www3.dsi.uminho.pt/pcortez/student.pdf

1. Retreiving data

the data is about student acheivement in secondary education of two Portugues schools according to the student grades, demographic and school-related features. Two datasets are available for the performance in two subjects: Math and Portuguese language. the datasets consists of 35 variables and 382 observations.I firstly beging with installing some packages helps analysis. And thus, we can take a glimpse at the datasets and what variables are available through column names on the datasets.

library(dplyr);library(ggplot2);library(tidyr);library(boot)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
alc<-read.csv("Z:\\IODS-project\\data\\alc.csv")
glimpse(alc)
## Observations: 382
## Variables: 36
## $ X          <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
## $ school     <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP,...
## $ sex        <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, ...
## $ age        <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15,...
## $ address    <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, ...
## $ famsize    <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, G...
## $ Pstatus    <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, ...
## $ Medu       <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, ...
## $ Fedu       <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, ...
## $ Mjob       <fct> at_home, at_home, at_home, health, other, services,...
## $ Fjob       <fct> teacher, other, other, services, other, other, othe...
## $ reason     <fct> course, course, other, home, home, reputation, home...
## $ nursery    <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, ye...
## $ internet   <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes,...
## $ guardian   <fct> mother, father, mother, mother, father, mother, mot...
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, ...
## $ studytime  <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, ...
## $ failures   <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ schoolsup  <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, ...
## $ famsup     <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes,...
## $ paid       <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, ...
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes,...
## $ higher     <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, y...
## $ romantic   <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no...
## $ famrel     <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, ...
## $ freetime   <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, ...
## $ goout      <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, ...
## $ Dalc       <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
## $ Walc       <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, ...
## $ health     <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, ...
## $ absences   <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6...
## $ G1         <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, ...
## $ G2         <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, ...
## $ G3         <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11,...
## $ alc_use    <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1...
## $ high_use   <lgl> TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE, TR...
gather(alc)%>% glimpse
## Warning: attributes are not identical across measure variables;
## they will be dropped
## Observations: 13,752
## Variables: 2
## $ key   <chr> "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "...
## $ value <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11",...
gather(alc)%>%ggplot(aes(value))+facet_wrap("key", scales="free")+geom_bar()
## Warning: attributes are not identical across measure variables;
## they will be dropped

2. Hypothesis

In order to study the relationship between high and low alcohol consumptions and variables, i am going to work folloing to hypothesis;

  1. Gender - Males consume more alcohol than females.
  2. G1 - First period grade decreases with alcohol consumption.
  3. G2 - Second period grade decreases with alcohol consumption.
  4. G3 - Third period grade decreases with alcohol consumption.

I will take a look at each relationship followring to the hypothesis one by one p<-ggpairs(alc, mapping = aes(col=sex, alpha = 0.3), lower = list(combo = wrap(“facethist”, bins = 20))) p

Hypothesis 1 - Gender

g1 <- ggplot(data = alc, aes(x = alc_use, fill = sex))
g1 + geom_bar()+facet_wrap("sex") + ggtitle("alohol consumption by sex")

g11<-ggplot(data=alc, aes(x=high_use, fill=sex))
g11+geom_bar()+facet_wrap("sex") + ggtitle("high and low alcohol consumption by sex")

Hypothesis 2 - G1

g2 <- ggplot(data=alc, aes(x=high_use, y= G1, col=sex))
g2+geom_boxplot()+ggtitle("G1 by alcohol consumption and sex")

Hypothesis 3 - G2

g3<-ggplot(alc, aes(x=high_use, y=G2, col=sex))
g3+geom_boxplot()+ggtitle("G2 by acohol consumption and sex")

Hypothesis 4 - Absences

g4<-ggplot(alc, aes(x=high_use, y=G3, col=sex))
g4+geom_boxplot()+ggtitle("G3 by alcohol consumption and sex")

Interim result

As a result of exploration of relationship with 4 variable to the high and low alcohol relationship, it seems that femailes consume more alcohol than males do. However, it is presumed that the more males consum alcohol, the less grades males achieve.

3.Lostistic regression

From now, in order to interpret relationship between variables presumably relating to alcohol consumptions and alcohol consumption, we will deal with logistaic regression.

m<-glm(high_use~sex+G1+G2+G3, data=alc, family="binomial")
summary(m)
## 
## Call:
## glm(formula = high_use ~ sex + G1 + G2 + G3, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.7050  -1.1393   0.8236   1.0275   1.4369  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -0.124551   0.377648  -0.330    0.742    
## sexM        -0.929752   0.215442  -4.316 1.59e-05 ***
## G1           0.100421   0.063556   1.580    0.114    
## G2           0.005342   0.076244   0.070    0.944    
## G3          -0.032884   0.053600  -0.614    0.540    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 524.94  on 381  degrees of freedom
## Residual deviance: 502.43  on 377  degrees of freedom
## AIC: 512.43
## 
## Number of Fisher Scoring iterations: 4
coef(m)
##  (Intercept)         sexM           G1           G2           G3 
## -0.124550963 -0.929752246  0.100420779  0.005341903 -0.032884237
OR<-coef(m)%>%exp
OR
## (Intercept)        sexM          G1          G2          G3 
##   0.8828933   0.3946515   1.1056360   1.0053562   0.9676506
CI<-exp(confint(m))
## Waiting for profiling to be done...
CI
##                 2.5 %    97.5 %
## (Intercept) 0.4202206 1.8520181
## sexM        0.2575779 0.5999088
## G1          0.9774251 1.2553494
## G2          0.8649561 1.1681070
## G3          0.8686826 1.0737634
cbind(OR, CI)
##                    OR     2.5 %    97.5 %
## (Intercept) 0.8828933 0.4202206 1.8520181
## sexM        0.3946515 0.2575779 0.5999088
## G1          1.1056360 0.9774251 1.2553494
## G2          1.0053562 0.8649561 1.1681070
## G3          0.9676506 0.8686826 1.0737634

The model indicates taht males are more significantly related to high alcohol consumptions.However, the other 3 variables are not significantly related to high alcohol consumptions. In odds rates, the odds rates for G2 and G3 are more than 1.

Prediction power

probabilities<-predict(m, type="response")
alc<-mutate(alc,probability=probabilities)
alc<-mutate(alc,prediction=probability>0.5)
select(alc,G2,G3, high_use, probability, prediction)%>% tail(10)
##     G2 G3 high_use probability prediction
## 373  5  0    FALSE   0.3953044      FALSE
## 374  5  5    FALSE   0.3567518      FALSE
## 375  9 10    FALSE   0.6454027       TRUE
## 376  5  6    FALSE   0.5762451       TRUE
## 377  5  0     TRUE   0.6468232       TRUE
## 378  9  8     TRUE   0.5898627       TRUE
## 379  5  0     TRUE   0.6235593       TRUE
## 380  5  0     TRUE   0.6235593       TRUE
## 381 16 16    FALSE   0.4777423      FALSE
## 382 12 10    FALSE   0.4466009      FALSE

2x2 cross-tabulation and graphic representation

g<-ggplot(alc,(aes(x=probability, y=high_use, col=prediction)))
g+geom_point()

table(high_use=alc$high_use, prediction=alc$prediction)%>%prop.table()%>%addmargins()
##         prediction
## high_use     FALSE      TRUE       Sum
##    FALSE 0.2225131 0.2225131 0.4450262
##    TRUE  0.1492147 0.4057592 0.5549738
##    Sum   0.3717277 0.6282723 1.0000000

Exploring the training error

loss_func<-function(class, prob){
  n_wrong <-abs(class - prob) > 0.5
  mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.3717277

Through exploring the training error, the incorrection of prediction is almost 50%.

10-fold cross-validation (Bonus task)

cv<-cv.glm(data=alc, cost=loss_func, glmfit = m, K=10)
cv$delta[1]
## [1] 0.395288

Eventually, the prediction error with the 4 variables is 40%.


Clustering and classification (Exercise 4)

library(MASS);library(tidyverse);library(corrplot)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
## -- Attaching packages --------------------------------------- tidyverse 1.2.1 --
## v tibble  2.1.3     v purrr   0.3.3
## v readr   1.3.1     v stringr 1.4.0
## v tibble  2.1.3     v forcats 0.4.0
## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()
## x MASS::select()  masks dplyr::select()
## corrplot 0.84 loaded
data("Boston")

The data has 506 observations and 14 variables. Data is about housing values in suburbs of Boston.

Explort dataset

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
pairs(Boston)

Correlate all variables

cor_matrix<-cor(Boston)
cor_matrix%>%round(2)
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)

There is strongly positive correlation bewtween “rad” and “tax” and stongly negative correlation between “age” and “dis” and between “lstat” and “medv”.

Standardized data

boston_scaled<-scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865

Covert matrix data to dafaframe

class(boston_scaled)
## [1] "matrix"
boston_scaled<-as.data.frame(boston_scaled)

Create a catergorical variable with the variable “Crime”

summary("crim")
##    Length     Class      Mode 
##         1 character character
bins<-quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
crime<-cut(boston_scaled$crim, breaks=bins, include.lowest=TRUE, label=c("low","med_low","med_high", "high"))
crime
##   [1] low      low      low      low      low      low      med_low 
##   [8] med_low  med_low  med_low  med_low  med_low  med_low  med_high
##  [15] med_high med_high med_high med_high med_high med_high med_high
##  [22] med_high med_high med_high med_high med_high med_high med_high
##  [29] med_high med_high med_high med_high med_high med_high med_high
##  [36] low      med_low  low      med_low  low      low      med_low 
##  [43] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
##  [50] med_low  med_low  low      low      low      low      low     
##  [57] low      low      med_low  med_low  med_low  med_low  med_low 
##  [64] med_low  low      low      low      low      med_low  med_low 
##  [71] med_low  med_low  med_low  med_low  low      med_low  med_low 
##  [78] med_low  low      med_low  low      low      low      low     
##  [85] low      low      low      low      low      low      low     
##  [92] low      low      low      low      med_low  med_low  med_low 
##  [99] low      low      med_low  med_low  med_low  med_low  med_low 
## [106] med_low  med_low  med_low  med_low  med_high med_low  med_low 
## [113] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [120] med_low  low      low      med_low  med_low  med_low  med_low 
## [127] med_high med_high med_high med_high med_high med_high med_high
## [134] med_high med_high med_high med_high med_high med_low  med_high
## [141] med_high med_high med_high high     med_high med_high med_high
## [148] med_high med_high med_high med_high med_high med_high med_high
## [155] med_high med_high med_high med_high med_high med_high med_high
## [162] med_high med_high med_high med_high med_high med_high med_high
## [169] med_high med_high med_high med_high med_low  med_low  med_low 
## [176] low      low      low      low      low      low      low     
## [183] med_low  med_low  med_low  low      low      low      med_low 
## [190] med_low  med_low  low      med_low  low      low      low     
## [197] low      low      low      low      low      low      low     
## [204] low      low      med_low  med_low  med_low  med_low  med_high
## [211] med_low  med_high med_low  med_low  med_high med_low  low     
## [218] low      med_low  med_low  med_high med_high med_high med_high
## [225] med_high med_high med_high med_high med_high med_high med_high
## [232] med_high med_high med_high med_high med_high med_high med_high
## [239] med_low  med_low  med_low  med_low  med_low  med_low  med_low 
## [246] med_low  med_high med_low  med_low  med_low  med_low  med_low 
## [253] med_low  med_high low      low      low      med_high med_high
## [260] med_high med_high med_high med_high med_high med_high med_high
## [267] med_high med_high med_high med_low  med_high med_low  med_low 
## [274] med_low  low      med_low  med_low  low      low      med_low 
## [281] low      low      low      low      low      low      low     
## [288] low      low      low      low      low      low      med_low 
## [295] low      med_low  low      med_low  low      low      low     
## [302] low      med_low  med_low  low      low      low      low     
## [309] med_high med_high med_high med_high med_high med_high med_high
## [316] med_low  med_high med_low  med_high med_high med_low  med_low 
## [323] med_high med_high med_high med_low  med_high med_low  low     
## [330] low      low      low      low      low      low      low     
## [337] low      low      low      low      low      low      low     
## [344] low      low      low      low      low      low      low     
## [351] low      low      low      low      low      med_low  high    
## [358] high     high     high     high     high     high     high    
## [365] med_high high     high     high     high     high     high    
## [372] high     high     high     high     high     high     high    
## [379] high     high     high     high     high     high     high    
## [386] high     high     high     high     high     high     high    
## [393] high     high     high     high     high     high     high    
## [400] high     high     high     high     high     high     high    
## [407] high     high     high     high     high     high     high    
## [414] high     high     high     high     high     high     high    
## [421] high     high     high     high     high     high     high    
## [428] high     high     high     high     high     high     high    
## [435] high     high     high     high     high     high     high    
## [442] high     high     high     high     high     high     high    
## [449] high     high     high     high     high     high     high    
## [456] high     high     high     high     high     high     high    
## [463] high     high     high     med_high high     high     high    
## [470] high     high     high     med_high high     high     high    
## [477] high     high     high     high     high     high     high    
## [484] med_high med_high med_high high     high     med_low  med_low 
## [491] med_low  med_low  med_low  med_low  med_high med_low  med_high
## [498] med_high med_low  med_low  med_low  low      low      low     
## [505] med_low  low     
## Levels: low med_low med_high high
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127

Remove original “crim” from the dataset and add the new categorial variable

boston_scaled<-dplyr::select(boston_scaled,-crim)
boston_scaled
##              zn       indus       chas         nox           rm
## 1    0.28454827 -1.28663623 -0.2723291 -0.14407485  0.413262920
## 2   -0.48724019 -0.59279438 -0.2723291 -0.73953036  0.194082387
## 3   -0.48724019 -0.59279438 -0.2723291 -0.73953036  1.281445551
## 4   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.015297761
## 5   -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.227362043
## 6   -0.48724019 -1.30558569 -0.2723291 -0.83445805  0.206891639
## 7    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9    0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11   0.04872402 -0.47618230 -0.2723291 -0.26489191  0.131459378
## 12   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13   0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.299402903
## 30  -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.554164692
## 31  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35  -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39  -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40   2.72854505 -1.19334657 -0.2723291 -1.09335175  0.441727925
## 41   2.72854505 -1.19334657 -0.2723291 -1.09335175  1.052302267
## 42  -0.48724019 -0.61611679 -0.2723291 -0.92075595  0.690796712
## 43  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50  -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53   0.41317968 -0.80123846 -0.2723291 -0.99842406  0.322174907
## 54   0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55   2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56   3.37170210 -1.44552019 -0.2723291 -1.30909650  1.372533565
## 57   3.15731641 -1.51548743 -0.2723291 -1.24868797  0.139998879
## 58   3.80047346 -1.43094368 -0.2723291 -1.24005818  0.756266222
## 59   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62   0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.243896145
## 64   0.58468822 -0.87557866 -0.2723291 -0.87760700  0.679410711
## 65   0.26310970 -1.42219777 -0.2723291 -1.19604625  1.166162284
## 66   2.94293073 -1.13212523 -0.2723291 -1.35224545  0.007636609
## 67   2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70   0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71  -0.48724019 -0.04763292 -0.2723291 -1.22279860  0.188389387
## 72  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74  -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76  -0.48724019  0.24681257 -0.2723291 -1.01568364  0.001943608
## 77  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80  -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.629596953
## 82   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.475885930
## 83   0.58468822 -0.91493524 -0.2723291 -1.11061133  0.024715612
## 84   0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85  -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.148538381
## 86  -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.491541682
## 87  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88  -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89  -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.028107013
## 90  -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.130581029
## 91  -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.188389387
## 92  -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.171310384
## 93   0.71331963  0.56895343 -0.2723291 -0.78267931  0.223970642
## 94   0.71331963  0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95   0.71331963  0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96  -0.48724019 -1.20209248 -0.2723291 -0.94664532  0.484425431
## 97  -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.539598741
## 99  -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.185209437
## 100 -0.48724019 -1.20209248 -0.2723291 -0.94664532  1.610216351
## 101 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.629596953
## 102 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.706452465
## 103 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.171310384
## 104 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.269514649
## 110 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.48724019 -0.16424500 -0.2723291 -0.06640675  0.612517950
## 113 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.590128481
## 122 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.399412952
## 123 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.460612711
## 124 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.610053984
## 125 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.577319229
## 126 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.425031456
## 127 -0.48724019  2.11552109 -0.2723291  0.22700611 -0.955903786
## 128 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.842043769
## 129 -0.48724019  1.56744433 -0.2723291  0.59808708  0.208314890
## 130 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.921745781
## 131 -0.48724019  1.56744433 -0.2723291  0.59808708  0.246742645
## 132 -0.48724019  1.56744433 -0.2723291  0.59808708  0.058873617
## 133 -0.48724019  1.56744433 -0.2723291  0.59808708  0.124343127
## 134 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.658444491
## 135 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.750955755
## 136 -0.48724019  1.56744433 -0.2723291  0.59808708  0.071682869
## 137 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.487654465
## 138 -0.48724019  1.56744433 -0.2723291  0.59808708  0.241049645
## 139 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.608630733
## 140 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.190195170
## 141 -0.48724019  1.56744433 -0.2723291  0.59808708 -0.157460416
## 142 -0.48724019  1.56744433 -0.2723291  0.59808708 -1.801314413
## 143 -0.48724019  1.23072696  3.6647712  2.72964520 -1.254786331
## 144 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.162275067
## 145 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.966411438
## 146 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.220083425
## 147 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.934555033
## 148 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.933676683
## 149 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.563631627
## 150 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.978675789
## 151 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.231469427
## 152 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.253363081
## 153 -0.48724019  1.23072696  3.6647712  2.72964520 -1.811277165
## 154 -0.48724019  1.23072696 -0.2723291  2.72964520 -0.819271765
## 155 -0.48724019  1.23072696  3.6647712  2.72964520 -0.221506675
## 156 -0.48724019  1.23072696  3.6647712  2.72964520 -0.188771920
## 157 -0.48724019  1.23072696 -0.2723291  2.72964520 -1.441232109
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## 381 -0.48724019  1.01499462 -0.2723291  1.00368721  0.972600255
## 382 -0.48724019  1.01499462 -0.2723291  1.00368721  0.370565414
## 383 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.065494052
## 384 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.088266056
## 385 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.727850303
## 386 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.434115858
## 387 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.323647242
## 388 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.828356167
## 389 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.999146193
## 390 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.273288584
## 391 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.813578764
## 392 -0.48724019  1.01499462 -0.2723291  1.25395112 -0.332520192
## 393 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.777119159
## 394 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.130418661
## 395 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.565933227
## 396 -0.48724019  1.01499462 -0.2723291  1.19354259  0.265244898
## 397 -0.48724019  1.01499462 -0.2723291  1.19354259  0.171310384
## 398 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.765188257
## 399 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.183623820
## 400 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.615746985
## 401 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.423608206
## 402 -0.48724019  1.01499462 -0.2723291  1.19354259  0.083068871
## 403 -0.48724019  1.01499462 -0.2723291  1.19354259  0.169887134
## 404 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.331641842
## 405 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.072610303
## 406 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.856276271
## 407 -0.48724019  1.01499462 -0.2723291  0.90012973 -3.055197852
## 408 -0.48724019  1.01499462 -0.2723291  0.90012973 -0.963020037
## 409 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.950210785
## 410 -0.48724019  1.01499462 -0.2723291  0.36508275  0.807503230
## 411 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.750955755
## 412 -0.48724019  1.01499462 -0.2723291  0.36508275  0.529969438
## 413 -0.48724019  1.01499462 -0.2723291  0.36508275 -2.357805247
## 414 -0.48724019  1.01499462 -0.2723291  0.36508275 -1.607752384
## 415 -0.48724019  1.01499462 -0.2723291  1.19354259 -2.512939520
## 416 -0.48724019  1.01499462 -0.2723291  1.07272553  0.212584640
## 417 -0.48724019  1.01499462 -0.2723291  1.07272553  0.707875715
## 418 -0.48724019  1.01499462 -0.2723291  1.07272553 -1.395688102
## 419 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.466305712
## 420 -0.48724019  1.01499462 -0.2723291  1.40928733  0.767652224
## 421 -0.48724019  1.01499462 -0.2723291  1.40928733  0.179849885
## 422 -0.48724019  1.01499462 -0.2723291  1.40928733 -0.396566452
## 423 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.906090028
## 424 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.258511181
## 425 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.024219796
## 426 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.553123975
## 427 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.637095738
## 428 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.117609410
## 429 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.130418661
## 430 -0.48724019  1.01499462 -0.2723291  1.07272553  0.135729129
## 431 -0.48724019  1.01499462 -0.2723291  0.25289548  0.090185122
## 432 -0.48724019  1.01499462 -0.2723291  0.25289548  0.780461476
## 433 -0.48724019  1.01499462 -0.2723291  0.25289548  0.199775388
## 434 -0.48724019  1.01499462 -0.2723291  1.36613839  0.215431141
## 435 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.109069908
## 436 -0.48724019  1.01499462 -0.2723291  1.59914271  0.490118432
## 437 -0.48724019  1.01499462 -0.2723291  1.59914271  0.251012396
## 438 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.188771920
## 439 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.497617217
## 440 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.935978283
## 441 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.664137492
## 442 -0.48724019  1.01499462 -0.2723291  1.59914271  0.172733634
## 443 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.093414156
## 444 -0.48724019  1.01499462 -0.2723291  1.59914271  0.285170401
## 445 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.612900484
## 446 -0.48724019  1.01499462 -0.2723291  1.59914271  0.248165896
## 447 -0.48724019  1.01499462 -0.2723291  1.59914271  0.080222370
## 448 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.047870149
## 449 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.141804663
## 450 -0.48724019  1.01499462 -0.2723291  1.36613839  0.188389387
## 451 -0.48724019  1.01499462 -0.2723291  1.36613839  0.660908458
## 452 -0.48724019  1.01499462 -0.2723291  1.36613839  0.527122938
## 453 -0.48724019  1.01499462 -0.2723291  1.36613839  0.017599361
## 454 -0.48724019  1.01499462 -0.2723291  1.36613839  1.577481596
## 455 -0.48724019  1.01499462 -0.2723291  1.36613839  0.631020203
## 456 -0.48724019  1.01499462 -0.2723291  1.36613839  0.342100410
## 457 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.439263958
## 458 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.496193967
## 459 -0.48724019  1.01499462 -0.2723291  1.36613839  0.023292362
## 460 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.289822685
## 461 -0.48724019  1.01499462 -0.2723291  1.36613839  0.592592447
## 462 -0.48724019  1.01499462 -0.2723291  1.36613839  0.130036128
## 463 -0.48724019  1.01499462 -0.2723291  1.36613839  0.046064365
## 464 -0.48724019  1.01499462 -0.2723291  1.36613839  0.325021407
## 465 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.107646658
## 466 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.748109254
## 467 -0.48724019  1.01499462 -0.2723291  0.86561057 -0.473421963
## 468 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.400836202
## 469 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.510426469
## 470 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.813578764
## 471 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.167423167
## 472 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.48724019  1.01499462 -0.2723291  0.21837632  0.216854391
## 474 -0.48724019  1.01499462 -0.2723291  0.51178918  0.989679257
## 475 -0.48724019  1.01499462 -0.2723291  0.25289548 -1.220628326
## 476 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.174539418
## 477 -0.48724019  1.01499462 -0.2723291  0.51178918  0.283747151
## 478 -0.48724019  1.01499462 -0.2723291  0.51178918 -1.395688102
## 479 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.141804663
## 480 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.079181654
## 481 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 -0.48724019  1.01499462 -0.2723291 -0.19585359  0.662331708
## 483 -0.48724019  1.01499462 -0.2723291 -0.19585359  1.104962525
## 484 -0.48724019  1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.588705230
## 486 -0.48724019  1.01499462 -0.2723291  0.24426569  0.038948114
## 487 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.242855428
## 488 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.540314723
## 489 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.182200570
## 490 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.239130578
## 491 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.695993897
## 492 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 493 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 494 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.822118266
## 495 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.510426469
## 496 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.874778524
## 497 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.273288584
## 498 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.698295497
## 499 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.378064199
## 500 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.018526795
## 501 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.366678197
## 502 -0.48724019  0.11562398 -0.2723291  0.15796779  0.438881424
## 503 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.234315927
## 504 -0.48724019  0.11562398 -0.2723291  0.15796779  0.983986256
## 505 -0.48724019  0.11562398 -0.2723291  0.15796779  0.724954717
## 506 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.362408446
##              age           dis        rad         tax     ptratio
## 1   -0.119894767  0.1400749840 -0.9818712 -0.66594918 -1.45755797
## 2    0.366803426  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 3   -0.265548971  0.5566090496 -0.8670245 -0.98635338 -0.30279450
## 4   -0.809087830  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 5   -0.510674339  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 6   -0.350809969  1.0766711351 -0.7521778 -1.10502160  0.11292035
## 7   -0.070159185  0.8384142195 -0.5224844 -0.57694801 -1.50374851
## 8    0.977840575  1.0236248974 -0.5224844 -0.57694801 -1.50374851
## 9    1.116389695  1.0861216287 -0.5224844 -0.57694801 -1.50374851
## 10   0.615481336  1.3283202075 -0.5224844 -0.57694801 -1.50374851
## 11   0.913894827  1.2117799501 -0.5224844 -0.57694801 -1.50374851
## 12   0.508905089  1.1547920492 -0.5224844 -0.57694801 -1.50374851
## 13  -1.050660656  0.7863652700 -0.5224844 -0.57694801 -1.50374851
## 14  -0.240681180  0.4333252240 -0.6373311 -0.60068166  1.17530274
## 15   0.565745754  0.3166899868 -0.6373311 -0.60068166  1.17530274
## 16  -0.428965883  0.3341187865 -0.6373311 -0.60068166  1.17530274
## 17  -1.395257187  0.3341187865 -0.6373311 -0.60068166  1.17530274
## 18   0.466274590  0.2198105553 -0.6373311 -0.60068166  1.17530274
## 19  -1.135921653  0.0006920764 -0.6373311 -0.60068166  1.17530274
## 20   0.032864520  0.0006920764 -0.6373311 -0.60068166  1.17530274
## 21   1.048891406  0.0013569352 -0.6373311 -0.60068166  1.17530274
## 22   0.732715207  0.1031753182 -0.6373311 -0.60068166  1.17530274
## 23   0.821528746  0.0863638874 -0.6373311 -0.60068166  1.17530274
## 24   1.116389695  0.1425444597 -0.6373311 -0.60068166  1.17530274
## 25   0.906789743  0.2871037683 -0.6373311 -0.60068166  1.17530274
## 26   0.608376252  0.3132232229 -0.6373311 -0.60068166  1.17530274
## 27   0.771793164  0.4212152950 -0.6373311 -0.60068166  1.17530274
## 28   0.718505041  0.3126533438 -0.6373311 -0.60068166  1.17530274
## 29   0.917447368  0.3132707128 -0.6373311 -0.60068166  1.17530274
## 30   0.665216917  0.2108349609 -0.6373311 -0.60068166  1.17530274
## 31   0.906789743  0.2079855659 -0.6373311 -0.60068166  1.17530274
## 32   1.116389695  0.1804414138 -0.6373311 -0.60068166  1.17530274
## 33   0.476932215  0.0925850666 -0.6373311 -0.60068166  1.17530274
## 34   0.938762618 -0.0037244859 -0.6373311 -0.60068166  1.17530274
## 35   1.006260907 -0.0167367233 -0.6373311 -0.60068166  1.17530274
## 36  -0.013318520 -0.2064589434 -0.5224844 -0.76681717  0.34387304
## 37  -0.254891346 -0.1981007179 -0.5224844 -0.76681717  0.34387304
## 38  -0.961847117  0.0660856927 -0.5224844 -0.76681717  0.34387304
## 39  -1.363284313  0.0248169544 -0.5224844 -0.76681717  0.34387304
## 40  -1.661697804  0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 41  -1.874850298  0.7627152911 -0.7521778 -0.92701927 -0.07184181
## 42  -2.333128159  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 43  -2.201684121  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 44  -2.205236663  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 45  -1.015135240  0.9145880470 -0.7521778 -1.03975408 -0.25660396
## 46  -1.235392817  0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 47  -1.253155525  0.6199131095 -0.7521778 -1.03975408 -0.25660396
## 48   0.601271169  0.8996287230 -0.7521778 -1.03975408 -0.25660396
## 49   0.949420242  0.9853955139 -0.7521778 -1.03975408 -0.25660396
## 50  -0.233576097  1.0887810641 -0.7521778 -1.03975408 -0.25660396
## 51  -0.812640371  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 52  -0.198050682  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 53  -1.686565595  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 54  -1.675907970  1.4340327636 -0.6373311 -0.98041997 -0.76469989
## 55  -0.745142082  1.6738568465 -0.7521778  0.36053095  1.22149328
## 56  -1.658145262  2.3277455193 -0.5224844 -1.08128796 -0.25660396
## 57  -1.167894527  2.5609210138 -0.8670245 -0.56508119 -0.53374719
## 58  -0.997372532  2.1511780064 -0.5224844 -0.90328562 -1.54993904
## 59  -1.398809729  1.9089794276 -0.1779443 -0.73715011  0.57482574
## 60  -0.759352248  1.4897384367 -0.1779443 -0.73715011  0.57482574
## 61  -0.084369352  1.6290738544 -0.1779443 -0.73715011  0.57482574
## 62   0.881921953  1.4358373805 -0.1779443 -0.73715011  0.57482574
## 63  -0.027528687  1.6291213443 -0.1779443 -0.73715011  0.57482574
## 64  -0.894348827  1.9878601804 -0.1779443 -0.73715011  0.57482574
## 65  -0.322389636  2.5776849546 -0.7521778 -1.14062207  0.06672981
## 66  -1.803799466  1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 67  -1.331311439  1.3375332514 -0.6373311 -0.42267932 -1.08803366
## 68  -1.675907970  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 69  -1.128816570  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 70  -1.263813149  1.2836321952 -0.6373311 -0.37521203  0.20530143
## 71  -2.201684121  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 72  -1.814457091  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 73  -2.159053622  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 74  -2.215894287  0.7086717651 -0.6373311 -0.61254848  0.34387304
## 75  -2.222999370  0.2167712006 -0.5224844 -0.06074124  0.11292035
## 76  -0.837508162  0.3360183832 -0.5224844 -0.06074124  0.11292035
## 77   0.210491598  0.1221237952 -0.5224844 -0.06074124  0.11292035
## 78  -0.809087830  0.1403124336 -0.5224844 -0.06074124  0.11292035
## 79  -0.528437047  0.5789293108 -0.5224844 -0.06074124  0.11292035
## 80  -1.135921653  0.3360183832 -0.5224844 -0.06074124  0.11292035
## 81  -1.246050442  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 82   0.064837394  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 83  -1.292233482  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 84  -0.777114956  0.7625253315 -0.6373311 -0.75495035  0.25149196
## 85  -0.730931915  0.4674704746 -0.7521778 -0.95668632  0.02053927
## 86  -0.443176049  0.3051974268 -0.7521778 -0.95668632  0.02053927
## 87  -0.833955621  0.3002109855 -0.7521778 -0.95668632  0.02053927
## 88  -0.418308258 -0.0225304932 -0.7521778 -0.95668632  0.02053927
## 89   0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450
## 90  -0.194498140 -0.1807194081 -0.8670245 -0.82021787 -0.30279450
## 91  -0.087921893 -0.3337319220 -0.8670245 -0.82021787 -0.30279450
## 92   0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450
## 93  -0.531989588 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 94  -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234
## 95   0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234
## 96  -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 97   0.036417061 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 98   0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
## 99  -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342
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## 317  0.519562713  0.0966691995 -0.6373311 -0.61848189 -0.02565127
## 318  0.111020434  0.1123883621 -0.6373311 -0.61848189 -0.02565127
## 319 -0.048843936 -0.1246813056 -0.6373311 -0.61848189 -0.02565127
## 320 -0.347257427  0.0982363667 -0.6373311 -0.61848189 -0.02565127
## 321 -0.578172628  0.3539695720 -0.5224844 -0.71934988  0.52863520
## 322 -0.507121797  0.3539695720 -0.5224844 -0.71934988  0.52863520
## 323 -0.663433626  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 324  0.203386515  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 325 -1.011582699  0.4397838527 -0.5224844 -0.71934988  0.52863520
## 326 -1.913928255  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 327 -1.409467353  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 328 -0.883691203  0.7697437989 -0.5224844 -0.71934988  0.52863520
## 329 -1.519596142  0.6741465952 -0.6373311  0.12912791 -0.71850935
## 330 -1.825114716  0.6741465952 -0.6373311  0.12912791 -0.71850935
## 331 -1.292233482  0.9871051509 -0.6373311  0.12912791 -0.71850935
## 332 -1.427230061  1.3514003073 -0.9818712 -0.61848189 -0.71850935
## 333 -1.608409681  1.3514003073 -0.9818712 -0.61848189 -0.71850935
## 334 -1.082633530  1.2648261879 -0.5224844 -1.09315478  0.80577843
## 335 -1.068423364  1.2648261879 -0.5224844 -1.09315478  0.80577843
## 336 -1.210525026  1.0401513886 -0.5224844 -1.09315478  0.80577843
## 337 -0.791325122  0.6819824315 -0.5224844 -1.09315478  0.80577843
## 338 -0.318837095  0.8642962245 -0.5224844 -1.09315478  0.80577843
## 339 -1.111053862  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 340 -0.823297996  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 341 -0.357915052  0.4830471675 -0.5224844 -1.09315478  0.80577843
## 342 -0.684748875  1.5400302593 -0.9818712 -0.73715011 -1.36517689
## 343 -0.315284553  1.1738829960 -0.9818712  0.08166062 -1.18041474
## 344 -0.432518424  0.9199069177 -0.5224844 -0.22687676 -0.39517558
## 345 -1.437887686  1.2681504821 -0.5224844 -0.22687676 -0.39517558
## 346 -0.713169208  2.0033893834 -0.7521778 -0.33367816  0.15911089
## 347 -0.578172628  2.0033893834 -0.7521778 -0.33367816  0.15911089
## 348 -1.452097852  2.2511442825 -0.6373311 -0.33961157 -0.25660396
## 349 -1.381047021  2.1602960705 -0.6373311 -0.76088376 -0.67231881
## 350 -1.210525026  2.3730983904 -0.9818712 -0.43454615  0.57482574
## 351 -0.858823412  2.3730983904 -0.9818712 -0.43454615  0.57482574
## 352 -1.160789444  3.2840499862 -0.6373311  0.01639310 -0.07184181
## 353 -1.778931675  3.2840499862 -0.6373311  0.01639310 -0.07184181
## 354 -1.153684361  3.9566021965 -0.5224844 -1.31269099 -0.67231881
## 355 -1.658145262  3.2248775491 -0.6373311 -0.44047956  1.63720813
## 356 -1.743406260  3.2248775491 -0.6373311 -0.44047956  1.63720813
## 357  1.024023615 -0.7944316108  1.6596029  1.52941294  0.80577843
## 358  0.796660955 -0.6125452271  1.6596029  1.52941294  0.80577843
## 359  0.526667797 -0.5092546567  1.6596029  1.52941294  0.80577843
## 360  0.452064424 -0.6106931203  1.6596029  1.52941294  0.80577843
## 361  0.690084708 -0.6063715378  1.6596029  1.52941294  0.80577843
## 362  0.800213497 -0.7121315839  1.6596029  1.52941294  0.80577843
## 363  0.981393116 -0.8032647354  1.6596029  1.52941294  0.80577843
## 364  0.725610124 -0.8977221812  1.6596029  1.52941294  0.80577843
## 365  0.508905089 -0.8977221812  1.6596029  1.52941294  0.80577843
## 366  0.686532167 -1.0361552904  1.6596029  1.52941294  0.80577843
## 367  0.810871121 -0.9700968153  1.6596029  1.52941294  0.80577843
## 368  1.116389695 -1.0848799457  1.6596029  1.52941294  0.80577843
## 369  1.116389695 -1.1694594886  1.6596029  1.52941294  0.80577843
## 370  1.002708366 -1.1579669285  1.6596029  1.52941294  0.80577843
## 371  1.027576156 -1.2312438711  1.6596029  1.52941294  0.80577843
## 372  1.116389695 -1.2470580136  1.6596029  1.52941294  0.80577843
## 373  0.746925373 -1.2658165310  1.6596029  1.52941294  0.80577843
## 374  1.116389695 -1.2446360278  1.6596029  1.52941294  0.80577843
## 375  1.116389695 -1.2623022771  1.6596029  1.52941294  0.80577843
## 376  1.041786323 -1.1771528552  1.6596029  1.52941294  0.80577843
## 377  0.878369411 -1.1635707388  1.6596029  1.52941294  0.80577843
## 378  1.073759197 -1.1573495596  1.6596029  1.52941294  0.80577843
## 379  0.981393116 -1.1440048928  1.6596029  1.52941294  0.80577843
## 380  1.116389695 -1.1440048928  1.6596029  1.52941294  0.80577843
## 381  0.828633829 -1.1295679579  1.6596029  1.52941294  0.80577843
## 382  1.084416821 -1.0807958128  1.6596029  1.52941294  0.80577843
## 383  1.116389695 -1.0517319833  1.6596029  1.52941294  0.80577843
## 384  1.116389695 -1.0741947142  1.6596029  1.52941294  0.80577843
## 385  0.803766038 -1.1186452769  1.6596029  1.52941294  0.80577843
## 386  1.048891406 -1.1250089259  1.6596029  1.52941294  0.80577843
## 387  1.116389695 -1.1054905698  1.6596029  1.52941294  0.80577843
## 388  0.743372832 -1.0811757321  1.6596029  1.52941294  0.80577843
## 389  1.116389695 -1.0474104008  1.6596029  1.52941294  0.80577843
## 390  1.077311738 -0.9815893753  1.6596029  1.52941294  0.80577843
## 391  1.009813449 -0.8873693792  1.6596029  1.52941294  0.80577843
## 392  0.494694923 -0.7727762084  1.6596029  1.52941294  0.80577843
## 393  1.009813449 -0.9616910999  1.6596029  1.52941294  0.80577843
## 394  0.853501620 -0.9516232374  1.6596029  1.52941294  0.80577843
## 395  0.928104993 -0.9559448199  1.6596029  1.52941294  0.80577843
## 396  1.073759197 -0.9827291333  1.6596029  1.52941294  0.80577843
## 397  0.974288033 -1.0059517029  1.6596029  1.52941294  0.80577843
## 398  1.077311738 -1.0265623271  1.6596029  1.52941294  0.80577843
## 399  1.116389695 -1.0948528283  1.6596029  1.52941294  0.80577843
## 400  0.327725469 -1.0897239172  1.6596029  1.52941294  0.80577843
## 401  1.116389695 -1.0477428302  1.6596029  1.52941294  0.80577843
## 402  1.116389695 -1.0547238481  1.6596029  1.52941294  0.80577843
## 403  1.116389695 -1.0239028917  1.6596029  1.52941294  0.80577843
## 404  0.974288033 -0.9936043244  1.6596029  1.52941294  0.80577843
## 405  0.597718628 -1.0389097056  1.6596029  1.52941294  0.80577843
## 406  1.116389695 -1.1253413553  1.6596029  1.52941294  0.80577843
## 407  1.116389695 -1.2427839210  1.6596029  1.52941294  0.80577843
## 408  1.116389695 -1.1919222195  1.6596029  1.52941294  0.80577843
## 409  1.041786323 -1.1114268095  1.6596029  1.52941294  0.80577843
## 410  1.116389695 -1.1062978984  1.6596029  1.52941294  0.80577843
## 411  1.116389695 -1.1312301050  1.6596029  1.52941294  0.80577843
## 412  1.116389695 -1.0768541496  1.6596029  1.52941294  0.80577843
## 413  1.116389695 -1.0643168114  1.6596029  1.52941294  0.80577843
## 414  1.116389695 -1.0474578907  1.6596029  1.52941294  0.80577843
## 415  1.116389695 -1.0147848276  1.6596029  1.52941294  0.80577843
## 416  1.116389695 -0.9309651233  1.6596029  1.52941294  0.80577843
## 417  0.789555872 -0.9381835908  1.6596029  1.52941294  0.80577843
## 418  0.729162665 -1.0198662487  1.6596029  1.52941294  0.80577843
## 419  1.116389695 -0.9462093868  1.6596029  1.52941294  0.80577843
## 420  0.281542429 -0.9502935197  1.6596029  1.52941294  0.80577843
## 421  1.116389695 -0.9194725633  1.6596029  1.52941294  0.80577843
## 422  0.949420242 -0.9120166463  1.6596029  1.52941294  0.80577843
## 423  0.675874542 -0.8756393696  1.6596029  1.52941294  0.80577843
## 424  0.587061003 -0.8421114879  1.6596029  1.52941294  0.80577843
## 425  0.071942477 -0.8223081923  1.6596029  1.52941294  0.80577843
## 426  0.952972784 -0.8953951752  1.6596029  1.52941294  0.80577843
## 427 -0.315284553 -0.8536040479  1.6596029  1.52941294  0.80577843
## 428  0.359698343 -0.9175729666  1.6596029  1.52941294  0.80577843
## 429  0.338383094 -0.8830477967  1.6596029  1.52941294  0.80577843
## 430  0.960077867 -0.8675660836  1.6596029  1.52941294  0.80577843
## 431  0.622586419 -0.8274371034  1.6596029  1.52941294  0.80577843
## 432  0.913894827 -0.8105781827  1.6596029  1.52941294  0.80577843
## 433  0.221149222 -0.7572944954  1.6596029  1.52941294  0.80577843
## 434  0.686532167 -0.7024911307  1.6596029  1.52941294  0.80577843
## 435  0.938762618 -0.7469416934  1.6596029  1.52941294  0.80577843
## 436  0.924552451 -0.7932443629  1.6596029  1.52941294  0.80577843
## 437  0.878369411 -0.8512295520  1.6596029  1.52941294  0.80577843
## 438  1.116389695 -0.8932106390  1.6596029  1.52941294  0.80577843
## 439  0.686532167 -0.9376612017  1.6596029  1.52941294  0.80577843
## 440  0.899684660 -0.9392758589  1.6596029  1.52941294  0.80577843
## 441  0.846396537 -0.9160057994  1.6596029  1.52941294  0.80577843
## 442  1.016918532 -0.8215483536  1.6596029  1.52941294  0.80577843
## 443  1.116389695 -0.8501847738  1.6596029  1.52941294  0.80577843
## 444  1.116389695 -0.8627221120  1.6596029  1.52941294  0.80577843
## 445  0.995603282 -0.9020437636  1.6596029  1.52941294  0.80577843
## 446  0.931657534 -0.8582105699  1.6596029  1.52941294  0.80577843
## 447  0.988498199 -0.8182715493  1.6596029  1.52941294  0.80577843
## 448  0.995603282 -0.7584342534  1.6596029  1.52941294  0.80577843
## 449  1.070206655 -0.7282306659  1.6596029  1.52941294  0.80577843
## 450  1.055996489 -0.7646079427  1.6596029  1.52941294  0.80577843
## 451  0.853501620 -0.6987869171  1.6596029  1.52941294  0.80577843
## 452  1.052443947 -0.6837801032  1.6596029  1.52941294  0.80577843
## 453  0.825081288 -0.6776064140  1.6596029  1.52941294  0.80577843
## 454  1.091521905 -0.6374774338  1.6596029  1.52941294  0.80577843
## 455  0.906789743 -0.6168668096  1.6596029  1.52941294  0.80577843
## 456  0.636796585 -0.6455032298  1.6596029  1.52941294  0.80577843
## 457  0.686532167 -0.5767378294  1.6596029  1.52941294  0.80577843
## 458  0.416539008 -0.4824228534  1.6596029  1.52941294  0.80577843
## 459  0.537325421 -0.4805707466  1.6596029  1.52941294  0.80577843
## 460  0.562193212 -0.5117241325  1.6596029  1.52941294  0.80577843
## 461  0.761135540 -0.5687120333  1.6596029  1.52941294  0.80577843
## 462  0.704294875 -0.5831489682  1.6596029  1.52941294  0.80577843
## 463  0.512457630 -0.5036983364  1.6596029  1.52941294  0.80577843
## 464  0.757582998 -0.4717851119  1.6596029  1.52941294  0.80577843
## 465 -0.112789684 -0.3949464255  1.6596029  1.52941294  0.80577843
## 466 -0.723826832 -0.3459843207  1.6596029  1.52941294  0.80577843
## 467  0.572850837 -0.4385896596  1.6596029  1.52941294  0.80577843
## 468  0.920999910 -0.5958762661  1.6596029  1.52941294  0.80577843
## 469  0.086152643 -0.4210658801  1.6596029  1.52941294  0.80577843
## 470 -0.421860800 -0.4612898402  1.6596029  1.52941294  0.80577843
## 471  0.547983046 -0.3617034834  1.6596029  1.52941294  0.80577843
## 472  0.786003330 -0.3304076278  1.6596029  1.52941294  0.80577843
## 473  0.228254306 -0.4267171803  1.6596029  1.52941294  0.80577843
## 474 -0.034633770 -0.5993905200  1.6596029  1.52941294  0.80577843
## 475  0.952972784 -0.6483526248  1.6596029  1.52941294  0.80577843
## 476  1.024023615 -0.7546350600  1.6596029  1.52941294  0.80577843
## 477  0.889027036 -0.7074775720  1.6596029  1.52941294  0.80577843
## 478  1.020471073 -0.8046419430  1.6596029  1.52941294  0.80577843
## 479  0.999155824 -0.7714939807  1.6596029  1.52941294  0.80577843
## 480  0.690084708 -0.8756393696  1.6596029  1.52941294  0.80577843
## 481 -0.137657475 -0.1761128861  1.6596029  1.52941294  0.80577843
## 482  0.224701764 -0.2200410597  1.6596029  1.52941294  0.80577843
## 483  0.299305137 -0.1825715149  1.6596029  1.52941294  0.80577843
## 484 -1.004477616  0.1440166471  1.6596029  1.52941294  0.80577843
## 485 -0.947636951 -0.0337381137  1.6596029  1.52941294  0.80577843
## 486 -0.592382795  0.0933923952  1.6596029  1.52941294  0.80577843
## 487  0.398776300 -0.1183176566  1.6596029  1.52941294  0.80577843
## 488 -0.546199754 -0.3052379716  1.6596029  1.52941294  0.80577843
## 489  0.857054162 -0.9375187319 -0.6373311  1.79641644  0.75958789
## 490  1.055996489 -0.9686246278 -0.6373311  1.79641644  0.75958789
## 491  1.045338864 -0.9367114033 -0.6373311  1.79641644  0.75958789
## 492  1.073759197 -0.9151034909 -0.6373311  1.79641644  0.75958789
## 493  0.530220338 -0.8002728706 -0.6373311  1.79641644  0.75958789
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512  0.34387304
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 497  0.153650933 -0.4732098094 -0.4076377 -0.10227512  0.34387304
## 498  0.071942477 -0.4285217972 -0.4076377 -0.10227512  0.34387304
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512  0.34387304
## 500  0.174966182 -0.6625521101 -0.4076377 -0.10227512  0.34387304
## 501  0.395223759 -0.6158695213 -0.4076377 -0.10227512  0.34387304
## 502  0.018654354 -0.6251775451 -0.9818712 -0.80241764  1.17530274
## 503  0.288647512 -0.7159307773 -0.9818712 -0.80241764  1.17530274
## 504  0.796660955 -0.7729186782 -0.9818712 -0.80241764  1.17530274
## 505  0.736267749 -0.6677760011 -0.9818712 -0.80241764  1.17530274
## 506  0.434301716 -0.6126402069 -0.9818712 -0.80241764  1.17530274
##            black        lstat         medv
## 1    0.440615895 -1.074498970  0.159527789
## 2    0.440615895 -0.491952525 -0.101423917
## 3    0.396035074 -1.207532413  1.322937477
## 4    0.415751408 -1.360170785  1.181588636
## 5    0.440615895 -1.025486649  1.486032293
## 6    0.410165113 -1.042290874  0.670558212
## 7    0.426376321 -0.031236706  0.039924923
## 8    0.440615895  0.909799859  0.496590409
## 9    0.328123258  2.419379350 -0.655946292
## 10   0.328999540  0.622727693 -0.394994586
## 11   0.392639483  1.091845624 -0.819041108
## 12   0.440615895  0.086392864 -0.394994586
## 13   0.370513376  0.428078760 -0.090550930
## 14   0.440615895 -0.615183504 -0.231899770
## 15   0.255720500 -0.335113097 -0.471105500
## 16   0.426595391 -0.585776111 -0.286264709
## 17   0.330533033 -0.850442645  0.061670899
## 18   0.329437681  0.282442149 -0.547216415
## 19  -0.741378303 -0.134862757 -0.253645746
## 20   0.375442459 -0.192277190 -0.471105500
## 21   0.217930861  1.171665689 -0.971262937
## 22   0.392749018  0.164812578 -0.318883672
## 23   0.440615895  0.849584722 -0.797295133
## 24   0.414765591  1.012025558 -0.873406047
## 25   0.412465352  0.510699530 -0.753803182
## 26  -0.583319029  0.540106923 -0.938643973
## 27   0.221326452  0.302047077 -0.645073304
## 28  -0.550896614  0.647934029 -0.840787084
## 29   0.342472368  0.020576319 -0.449359525
## 30   0.258020739 -0.094252548 -0.166661844
## 31   0.038293155  1.392921311 -1.069119826
## 32   0.219683424  0.054184768 -0.873406047
## 33  -1.359047220  2.108501199 -1.014754888
## 34   0.022958229  0.797771697 -1.025627875
## 35  -1.186967442  1.076441751 -0.982135924
## 36   0.440615895 -0.416333515 -0.394994586
## 37   0.228774844 -0.174072614 -0.275391721
## 38   0.440615895 -0.543765550 -0.166661844
## 39   0.402607185 -0.353317674  0.235638703
## 40   0.426704926 -1.166922204  0.898890955
## 41   0.426595391 -1.494604580  1.344683452
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boston_scaled<-data.frame(boston_scaled, crime)

Divide the dataset into train and test

n<-nrow(boston_scaled)
ind<-sample(n, size = n*0.8)
train<-boston_scaled[ind, ]
test<-boston_scaled[-ind, ]
correct_classes<-test$crime

Linear Discriminant Analysis

lda.fit<-lda(crime~., data=train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2623762 0.2549505 0.2301980 0.2524752 
## 
## Group means:
##                  zn      indus         chas        nox         rm
## low       0.9724027 -0.9417505 -0.123759247 -0.8833222  0.4306105
## med_low  -0.1560872 -0.2347076 -0.004759149 -0.5485858 -0.1737656
## med_high -0.3563036  0.1249185  0.108680633  0.2809191  0.2250725
## high     -0.4872402  1.0171096 -0.040734936  1.0370219 -0.3903153
##                 age        dis        rad        tax      ptratio
## low      -0.8921034  0.9075613 -0.6882537 -0.7457144 -0.420885405
## med_low  -0.2658939  0.3091336 -0.5581649 -0.4607568  0.003946553
## med_high  0.4027872 -0.3522652 -0.4014631 -0.3258307 -0.292861051
## high      0.8108363 -0.8626504  1.6382099  1.5141140  0.780871766
##               black       lstat        medv
## low       0.3753774 -0.75738529  0.53064543
## med_low   0.3096863 -0.06205805 -0.05117594
## med_high  0.1571859 -0.07448199  0.28953599
## high     -0.7872832  0.91360278 -0.69730628
## 
## Coefficients of linear discriminants:
##                  LD1         LD2         LD3
## zn       0.121325608  0.77207565 -0.75846084
## indus    0.053895373 -0.41181402  0.19835527
## chas    -0.008812182 -0.01759874  0.14678878
## nox      0.520596914 -0.57053926 -1.46183169
## rm       0.018117723 -0.06453497 -0.18151489
## age      0.197398037 -0.35101243 -0.06474741
## dis     -0.074070315 -0.31204251  0.09798344
## rad      3.149134426  0.90341071 -0.05691434
## tax     -0.049102223  0.02712580  0.60007210
## ptratio  0.090762409  0.08925563 -0.17598439
## black   -0.119527655 -0.00867947  0.10688625
## lstat    0.221627662 -0.27198749  0.29787702
## medv     0.090152036 -0.39838427 -0.30917956
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9497 0.0363 0.0140
lda.arrows<-function(x, myscale=1, arrow_heads= 0.1, color="red", tex=0.75, choices =c (1,2)){
  heads<-coef(x)
  arrows(x0 = 0, y0=0, x1 = myscale*heads[,choices[1]],y1=myscale*heads[,choices[2]], col=color, length= arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

Convert target classes to numeric data

classes<-as.numeric(train$crime)

Plot the lda results

plot(lda.fit, dimen=2, col=classes, pch=classes)
lda.arrows(lda.fit, myscale=1)

Predict the classes with the LDA model and Cross-tabulate the results

lda.pred<-predict(lda.fit, newdata = test)
table(correct=correct_classes, predict=lda.pred$class)
##           predict
## correct    low med_low med_high high
##   low       13       7        1    0
##   med_low    9      13        1    0
##   med_high   0      14       18    1
##   high       0       0        0   25

Distance measures

library(MASS)
data("Boston")
boston_scaled<-scale(Boston)
boston_scaled<-as.data.frame(boston_scaled)
Euclidean distane matrix
dist_eu<-dist(boston_scaled)
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

K-means

km<-kmeans(boston_scaled, centers=3)
pairs(boston_scaled, col=km$cluster)

Investigate the optimal number of clusters

set.seed(11)
k_max<-20
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
qqplot(x=1:k_max, y = twcss, geom="Line")
## Warning in plot.window(...): "geom" is not a graphical parameter
## Warning in plot.xy(xy, type, ...): "geom" is not a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter

## Warning in axis(side = side, at = at, labels = labels, ...): "geom" is not
## a graphical parameter
## Warning in box(...): "geom" is not a graphical parameter
## Warning in title(...): "geom" is not a graphical parameter

Perform k-means with different number of clusters

km<-kmeans(boston_scaled, centers=6)
pairs(boston_scaled, col = km$cluster)

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
## 
## Attaching package: 'GGally'
## The following object is masked from 'package:dplyr':
## 
##     nasa
ggpairs(boston_scaled)

Super Bonus

model_predictors<- dplyr::select(train, -crime)
Check th dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
Matrix multiplication
matrix_product <- as.matrix(model_predictors)%*%lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
Create a 3D plot
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers")
plot_ly(x=matrix_product$LD1, y=matrix_product$LD2, z=matrix_product$LD3, type="scatter3d", mode="markers", color = train$crime)
km <-kmeans(boston_scaled, centers = 6)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])